Facilitating productive mathematical argumentation, especially asking rational questions, is essential yet remains challenging for pre-service mathematics teachers (PMTs), who often have limited opportunities to apply abstract theoretical knowledge in authentic practice. At the same time, recent advances in large language models (LLMs) have expanded the potential for simulating students in educational settings, enabling low-risk environments for instructional practice. To inform the design of a system that supports PMTs in orchestrating classroom argumentation, we conducted a formative study with eight experienced mathematics teachers to identify key design requirements, including personalization, realistic simulations, structured reflection, and ease of use. Building on these requirements, we developed ArguMath, an AI-simulated classroom environment that supports PMTs in practicing the orchestration of mathematical argumentation. ArguMath comprises three core components: (1) customization of classroom settings; (2) simulation of classroom discussions with AI-based students grounded in authentic transcripts and augmented with real-time instructional suggestions; and (3) structured reflection through discourse annotation and overall feedback. Results from an exploratory user study with seven PMTs, complemented by interviews with four experienced teachers, indicate that ArguMath has the potential to support PMTs' classroom orchestration skills, particularly theory-aligned questioning strategies.

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